1. Prove that:(25p) [A x (Bx C)] + [Bx( Cx A)] + [Cx( Ax B)]=0 2. Three vectors A , B and C are given by: A = 3x-2y+22 B = 6x+4y-2z C=-3x-2y-42 Compute the values of A . Bx C and A x (Bx C) and Cx( Ax B) and Bx(CX A). (25p) 3. Show that a.( b+ c)= a. b+a. c in two dimensions. (Warning! This exercise mostly includes geometry and trigonometry). (25p) 4. Use Gram-Schmidt orthonormalization to show that the three vectors (2, - 1, 3), (-1,1, -2) and (3, 1, 2) are linearly dependent. (25p)( einstin's notation ) Axt Ay = AZ= 0 phythag crew -llecrem a times AZZ Ax Ay + AZ A = A COs a + A EOS B + AR COS 1 - Co's at ( OS 2 B + 105 2 8 (9 ) Formal definition of vector @e one boothing for x'sy cc . find x = x cost + y sing cost sink ) yo - - 7 sing + y cos$ sing cosop 2605 8 + 4 sind - xx sind + y sings Quiz - sink 0 21 = - sing azz = cost 1 = 412 ( Haydi bistoamela Imn yser aandi kleen vectors ofser complicated ) n The Generalization to U dimensions became Simply Vi = 2 aij Vi 1=1 al is he cosine of de angle bete 1 = 12 - V 4 x; one called director cosines+6 Derivativess Graders 8 One variables fix) is a scaler function df ( x ) = dfu dx same as fica - afew doc d xx Proportionality out Rate of change ( slap ) af( ) . 178 - proportionality facks beto the change by the argument + change in the function Lit tells too fast te change occurs 3 variables : (pix, y, 2 ) is a scaler function depends on the values of the coordinates ( x,y , ? ) we wants to generale the notion derivative to lunchin like op , which depend not ohone but A 3 variables . A derivative is supposed to tell how fast the function change if we move a little distance. But this time the situation depends on wheel direction we move . we need a vector operator that defins the change of a scales function in any direction since Day, 2) is a scaler function it must be independent of rotation y the coordinate system. ' ( 2, ( , x 2 / x(3 ) = CD ( x, , p( 2, x( 3 ) V ' = zaijVi Ox' vector transfermation law . The vecter has the component ad then we can define a veter operater that acts on scaler Vod da O dy ( sed V ( deloperater )15 Triple Vector produd & A X ( B X C ) + ( A x B ) X C x x ( x xy ) = 3c x 2 = _ g ( D C X 32 ) x y = 0 A X ( B X C ) : B X C = BIG EK OF = DK ek = D DK A X ( B X C) = A XD AmDK Eman en Ger = C ( nfs she. ) = Am Big Erjk Eman en - Am Bic ZijK Emaken = Am Bici ( I ( Sim ajn - sin SymDen AIBI CAen + Aj Bn gen - CA. BC + CA. CB = BA .C) _ C ( AB) (Rule = BAC - CAB 2 Geometric Meaning ? BXC B x BA C are in the xy plane BXC is perp to the ay plane Then RX ( B X ( ) is perpendicular to the z axis Ther fore is back in the Jay plane(A B ) . ( , A . ( B 2 ) = for bidden 14 ) Triple scaler products A . ( B X C ) = Ax ( By ( z . Bady ) + Ay ( Bzay - B xx ( 2 ) + AZ ( Bolly- By(2 mabish b x - B CX A C . A X B A. CX B - . B XA = - B. Axc handi bs byone change position fbtseer @ @bthis Iproud A . B X C = A K B . C Z j K BIAKG EK Binterchage = > BIAK C Erji - BICA XC ) ; - Bi (AXC) B. (2 B) A . B X C = A K BI G E jK GAK Bi SAK CAK BIZKy 9 ( A x B ) , CA X B 7. B X A A. B XC Ax Ay AZ - Ax ( By ( 2 - Bzay) - Ay ( Bx (2 BZ(x ) + AZ ( Bxly- By (x) BX By BZ Hayda shk/ parm dex bazil and Volum th 30 magnitude A . ( B X ( ) - volume of parallel piped defined by A B 42515. 1 AR 1 99 5, C.1 for B-Dimension Inthe plane of I vectors a, as we have already bound 2 orthonored vecters ese To Get the 3' vector we do the following decomposition 2 93 = 93 12+ 931 +93/1 perp to phee 9 3/1 - age costre - cas ever asH - a3 cos Ozle - ( as e ) e 2 a31 = 93 - 9311 - 9371 ash es - as- cap ese - (93 . ez Dez - 93 - cas . ese - caller De =al ata career 1102- ca ezell\f16 veder a vect spock DCL- ( 241 1 2( 2 , D(3 ) collecting such vectors form a vector space with the folkcoy properties . 1. vecter equality 2 vector addition 3 Scaler multiplicalu a's read 4. hegale & avader 5. hull vecher there exist a null vecher OF (0, 0) The to locky properties cuso hold : 1. 3* + 3 - 9+x (commutalivity) 8. (:4 ) 72 - 2+ ( 9 1 2 ) cassociativity 3. Scaler multiplicaha is dot butve a (7+g ) - astay ( atb ) 7 - ax tbxx 4. S calor multiplical is associative ( ab ) ? = a(bx ) 5 . hull wecker is unique 12 = 34 41_ 87 17 2x - y = 7: 46 -202all = 8x Quiz - 1024 aijt beil if we rakde los bahyer Is positice Lb3a The inverse relater ( co - () h -tensors obey this love . too Eajaik-- Sik of Zajlake - Sik some free 3 5 05 2)8 index in Rooth sides where SjK - 1 for j=* 5 1 fer ke j to fer k * j 8x ; Dejack are totally independent variables